Linear Inequalities

An inequality swaps the equals sign for <, >, ≤, or ≥ to compare two quantities. An equation has one answer, but an inequality is met by a whole range of numbers, like every height tall enough to ride. Here you shade that range on a number line and watch the sign flip when you multiply both sides by a negative.

Intuition — Inequalities Instead of Equality

👀 What is an Inequality?

①Equation: x = 3 → exactly one answer

②Inequality: x > 3 → every number greater than 3

③The solution is a range, not a single value

Boundary3

Sign (0:< 1:≤ 2:> 3:≥)2

Property — The Negative-Multiplication Trap

Multiplier2

Core Property

Multiplying/dividing both sides by a negative → flip the sign

Multiplying/dividing by a positive keeps the sign

Solving Linear Inequalities

Steps

Expand brackets → move terms → combine like terms → divide by coefficient

Same as solving a linear equation, but flip when dividing by a negative

📝 Worked Example

①Problem: 2x - 3 > 5

②Move: 2x > 5 + 3 = 8

③Divide by 2: x > 4

④On a number line: x > 4 (open circle at 4, arrow to the right)

Word Problems

💡 How to Approach Word Problems

①Look for "more than / less than / at least / at most" → choose sign

②Set a variable, write the inequality

③Solve, then keep solutions matching the conditions

④If "natural numbers" only, pick integer solutions in the range

Work It Out

Example 1

Solve the inequality 2x − 3 > 5.

Move the constant to the right.

2x > 5 + 3 = 8

Divide both sides by positive 2 (inequality unchanged).

x > 4

▸ x > 4

Dividing by a positive number keeps the inequality direction.

Example 2

Solve the inequality −3x + 1 ≥ 7.

Move the constant.

−3x ≥ 6

Dividing by negative −3 flips the inequality.

x ≤ −2

▸ x ≤ −2

Multiplying or dividing by a negative always flips the inequality.

Exam Wrap-up

Core Idea

ax + b > 0 → x > -b/a (when a > 0)

When a < 0 the sign flips!

Grade-8 school exam type

How many natural numbers x satisfy 5 − 2x ≥ −1?

①1

②2

③3

④4

⑤5

▸ ③ 3

From −2x ≥ −6, dividing by a negative flips the sign.

−2x ≥ −6 ⇒ x ≤ 3

Natural numbers with x ≤ 3 are 1, 2, 3.

x = 1, 2, 3 ⇒ 3 numbers

🎯 Exam Points

①Number line: <,> open circle (○); ≤,≥ filled circle (●)

②Multiplying/dividing by a negative flips the sign (most-missed!)

③Don't forget the sign change when moving terms

④"At least / at most" includes the boundary; "more / less than" excludes it

⑤If natural-number constraint, give only integers

← Previous

Polynomials

System of Equations

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